aa r X i v : . [ m a t h . A C ] J a n AUSLANDER’S THEOREM AND n -ISOLATED SINGULARITIES JOSH STANGLEABSTRACT. One of the most stunning results in the representation theory of Cohen-Macaulay rings isAuslander’s well known theorem which states a CM local ring of finite CM type can have at most an isolatedsingularity. There have been some generalizations of this in the direction of countable CM type by Huneke andLeuschke. In this paper, we focus on a different generalization by restricting the class of modules. Here weconsider modules which are high syzygies of MCM modules over non-commutative rings, exploiting the fact thatnon-commutative rings allow for finer homological behavior. We then generalize Auslander’s Theorem in thesetting of complete Gorenstein local domains by examining path algebras, which preserve finiteness of globaldimension.
1. Introduction.
One main focus of the study of representation theory of commutative Noetherian rings isthe question of finite Cohen-Macaulay (CM) type–i.e., when a local commutative Noetherian ring R has onlyfinitely many (up to isomorphism) indecomposable maximal Cohen-Macaulay modules. Auslander showedthat a complete Cohen-Macaulay local ring R of finite CM type has at most an isolated singularity; that is,gldim R p = dim R p for all non-maximal prime ideals p ∈ Spec R , [2]. Wiegand [23] and Leuschke-Wiegand [17] then proved thatfinite CM type ascends to and descends from the completion of an excellent local ring R , thus generalizing thetheorem to all excellent CM local rings. Finally, Huneke-Leuschke gave a completion-free proof for arbitraryCM local rings in [13]. In the paper of Huneke-Leuschke, the idea of countable CM type is addressed, andthey are able to show that if a CM local ring has countable CM type then the singular locus is at most one-dimensional. In this paper we are interested in a different generalization of Auslander’s theorem. We wish torestrict our finiteness assumption to a smaller class of modules. To do so we will consider non-commutativealgebras over which high-syzygies exhibit similar behavior to maximal Cohen-Macaulay modules over CMlocal rings. Such algebras will be called n -canonical orders, and Section 4 will focus on their properties.The main result, Theorem 5.1, is that for an n -canonical R -order Λ , if there are only finitely many (up toisomorphism) indecomposable n th syzygies of MCM Λ -modules, then gldim Λ p É n + dim R p for all non-maximalprime ideals p ∈ Spec R .Finally, in section 6 we refocus on the case of commutative rings, using our main theorem to show that if R isa complete Gorenstein local domain and Q is an acyclic quiver such that RQ has finitely many indecomposablefirst syzygies (of MCM RQ -modules), then R has at most an isolated singularity. This is a generalization ofAuslander’s theorem for Gorenstein domains. Received by the editors January 22, 2021. . Background and Notation. Here we will briefly remind the reader of the notation, conventions, anddefinitions which are heavily utilized in this article. Throughout, R will be a commutative Noetherian ring offinite Krull dimension d . We use the notation ( R , m , k ) to imply R is a commutative local Noetherian ring withmaximal ideal m and residue field R / m = k . For the convenience of the reader, below we include definitions ofthe preliminary notions we will use. Definition 2.1.
Let ( R , m , k ) be a commutative local Noetherian ring of finite Krull dimension d. • Let M be a finitely generated R-module. We say M is maximal Cohen-Macaulay (MCM) if depth( M ) : = min { i Ê | Ext iR ( k , M ) } = d . • An R-algebra Λ is an R- order if it is a MCM R-module. • Denote by
Mod Λ the category of left Λ -modules and mod Λ the full subcategory of Mod Λ consisting offinitely generated modules. Unless specified otherwise, when we say M is a Λ -module, we always meana finitely generated left Λ -module. • We denote by CM Λ the full subcategory of mod Λ consisting of modules which are maximal Cohen-Macaulay R-modules. • For a (possibly non-commutative) ring Γ , we will denote by Γ op the opposite ring. If M is an abeliangroup with a right Γ -module structure, we will say M ∈ mod Γ op to indicate that M is a left Γ op -module. • An R-order Λ is non-singular if gldim Λ p = dim R p for all p ∈ Supp R Λ . • We say an R-order Λ is an isolated singularity if gldim Λ p = dim R p for all non-maximal prime ideals p ∈ Supp R Λ . • For any ring Γ , we denote by proj Γ the full subcategory of mod Λ consisting of all projective Γ -modules. • For any module M, add
M denotes the additive closure of M, i.e., the full subcategory of
Mod Λ consistingof all modules which are isomorphic to direct summands of finite direct sums of copies of M. For details on Cohen-Macaulay rings or canonical modules, see [7, Section 3.3]; for details on orders, see[9, 21]; for details on module-finite algebras with injective dimension in mind, see [12].In the case that R is Cohen-Macaulay with a canonical module ω R , an R -order possesses a special moduleakin to ω R . Definition 2.2.
Let R be a Cohen-Macaulay ring with canonical module ω R and Λ an R-order. Then the canonical module of Λ is ω Λ = Hom R ( Λ , ω R ) . We see that ω Λ is both a Λ − and Λ op -module. It will be useful to note that the canonical module of an order over a Cohen-Macaulay local ring is invariantunder change of base ring. 2 emma 2.3.
Suppose R is a CM local ring with a canonical module ω R and that S , → R is ring extension suchthat R ∼= Hom S ( R , S ) (e.g., if R is a Gorenstein ring which is an order over a regular local ring S). Suppose Λ is an algebra which is both an S-order and an R-order. Then, the canonical modules of Λ as an R-order isisomorphic (as a Λ -module) to the canonical module of Λ as an S-order.Proof. All we need to establish is that Hom S ( Λ , S ) ∼= Hom R ( Λ , R ), i.e. that the canonical module of Λ as an R -order agrees with that as an S -order. We seeHom R ( Λ , R ) ∼= Hom R ( Λ ,Hom S ( R , S )) ∼= Hom S ( Λ ⊗ R R , S ) ∼= Hom S ( Λ , S ).It is straight-forward to verify this is also an isomorphism of Λ -modules. (cid:3) In the rest of this article, R is always assumed to be a Cohen-Macaulay local ring with canonical module ω R . Let Λ be an R -order. We have the following functors. • The canonical dual D d ( − ) : = Hom R ( − , ω R ) : CM Λ −→ CM Λ op . Note, this functor is exact on CM Λ sinceExt iR ( M , ω R ) = i > M an MCM R -module. • The
Matlis dual D : = Hom R ( − , E ) where E is the injective hull of the residue field, k , of R . Lettingf.l. R denote the full subcategory of mod R consisting of finite length R -modules, D : f.l. R −→ f.l. R is aduality. • The functor ( − ) ∗ : = Hom Λ ( − , Λ ) : mod Λ −→ mod Λ op which gives a duality ( − ) ∗ : add Λ −→ add Λ op . • The transpose duality
Tr : mod Λ −→ mod Λ op given by Tr M = cok f ∗ , where P f −→ P f −→ M −→ M . • Finally, we denote Hom R ( − , R ) = ( − ) † . In the case when R is Gorenstein, we note that D d ( − ) = ( − ) † .
3. Projective Dimension and the Canonical module.3.1. n -Canonical Orders. In this section we examine orders which exhibit similar behavior as seen incommutative rings. Specifically, we note that by the Auslander-Buchsbaum formula [4], maximal Cohen-Macaulay modules over commutative rings are either projective or have infinite projective dimension. Weprove that for orders over CM rings, finite projective dimension of the canonical modules gives a similar resultfor high syzygies. The central objects of this paper, n -canonical orders, are a generalization of the followingwell-studied class of orders. Definition 3.1.
Let R be a Cohen-Macaulay ring with canonical module ω R and Λ an R-order. If ω Λ isprojective as a left Λ -module, then Λ is called a Gorenstein order . R is a CM local ring–for noncommutative crepant resolutions. One reason that Gorensteinorders are so useful is that they exhibit some similar behavior to commutative rings. In particular, they satisfyan Auslander-Buschbaum theorem. Lemma 3.2. [15, Lemma 2.16]
Let Λ be a Gorenstein R-order. Then for any X ∈ mod Λ with projdim Λ X < ∞ we have projdim Λ X + depth R X = dim R .The above result is a special case of the main result of this section, which relates the projective dimensionof ω Λ to the possible projective dimension of all finitely generated Λ -modules.It is important to note that the Gorenstein property is defined to be one-sided. It is natural to ask if theproperty passes from Λ to Λ op . Remark 3.3.
The Gorenstein property is symmetric, as shown in [15, Lemma 2.15] : Λ is Gorenstein if and onlyif Λ op is Gorenstein. The rest of this paper is dedicated to orders with projdim Λ op ω Λ É n . As such, we give this condition a name. Definition 3.4.
Let R be a CM local ring with canonical module ω . Let Λ be an R-order. We call Λ n- canonical if projdim Λ op ω Λ É n. The above definition can be considered a one-sided version of the the n -Gorenstein property defined in[10, Section 9]. There, a left and right Noetherian ring Λ is called n - Gorenstein (alternatively,
Iwanaga-Gorenstein) if there is a non-negative integer n such that injdim Λ Λ É n and injdim Λ op Λ É n . If one defines left n - Gorenstein to mean only injdim Λ Λ É n , then the following proposition (an application of [11, Proposition1.1(3)]) yields that an n -canonical order Λ over a d -dimensional local Cohen-Macaulay ring R is precisely aleft ( n + d )-Gorenstein order. Proposition 3.5.
Let Λ be an order over a Cohen-Macualay local ring R of dimension d. The following areequivalent:(1) The Λ op -module ω Λ has projective dimension at most n.(2) The Λ -module Λ has injective dimension at most d + n. n -Gorenstein has another definition given by Auslander and Reiten in [6]. According to Auslanderand Reiten, a Noetherian ring Λ is called n -Gorenstein if flatdim Λ I i É i for 0 É i < n for a minimal injectiveresolution 0 −→ Λ −→ I −→ I −→ I −→ ··· .The relation between Auslander and Reiten’s definition of n -Gorenstein and the above definition ofIwanaga-Gorenstein has been well-studied; see, e.g., [6, 16].As in Remark 3.3, one may ask whether being n -canonical is a symmetric property. Remark 3.3 establishesthis in the affirmative when n =
0. It is further known to be true in the case n = Gorenstein symmetry conjecture and seems quite difficult. The following result followsfrom the work in [25], and will be useful for us.
Proposition 3.6.
Let Λ be an order over a Cohen-Macaulay local ring R. If both projdim Λ ω Λ and projdim Λ op ω Λ are finite, then Λ is n-canonical if and only if Λ op is n-canonical. In particular, if gldim Λ < ∞ ,then Λ is n-canonical if and only if Λ op is n-canonical.Proof. We observe that projdim Λ ω Λ = max { i Ê | Ext i Λ ( ω Λ , Λ ) } = max { i Ê | Ext i Λ op ( ω Λ , Λ ) } = projdim Λ op ω Λ .Here, the first and third inequalities follow since we know both projdim Λ ω Λ and projdim Λ op ω Λ are finite. Thesecond equality follows from the exact duality D d : CM( Λ ) −→ CM( Λ op ). (cid:3) Notation 3.7.
We define an n th - syzygy of a module X ∈ mod Λ , denoted Ω n X to be a module Y appearing inan exact sequence −→ Y −→ P n − −→ ··· −→ P −→ X −→ where the P i are finitely generated projective Λ -modules. Denote by X n the additive closure of the fullsubcategory of mod Λ consisting of all n th -syzygies of maximal Cohen-Macaulay Λ -modules, i.e., X n = add { Ω n CM Λ } = add { M ∈ mod Λ | M ∼= Ω n X for some X ∈ CM Λ } .Now we move on to prove the main result of this section, a generalized Auslander-Buchsbaum formula for n -canonical orders. It follows from Ext-vanishing imposed by being n -canonical.5 emma 3.8. Suppose Λ is an n-canonical order over a CM local ring R with a canonical module. If M ∈ CM Λ then Ext i Λ ( M , Λ ) = for i > n. In particular, if X ∈ X n , then Ext i Λ ( X , Λ ) = for i > .Proof. Begin by taking a projective resolution of ω Λ over Λ op −→ P n −→ P n − −→ ... −→ P −→ ω Λ −→ D d ( − ) to get a resolution0 −→ Λ −→ I −→ ... −→ I n − −→ I n −→ I j ∈ add ω Λ . Since Ext i Λ ( N , ω Λ ) = i > N ∈ CM Λ , we have Ext n + i Λ ( N , Λ ) =
0. The final statementthen follows by dimension shifting for Ext. (cid:3)
Theorem 3.9.
Suppose Λ is an n-canonical order over a CM local ring with a canonical module. For anyX ∈ mod Λ with projdim Λ X < ∞ we have (3.1) dim R É projdim Λ X + depth R X É dim R + n . Proof.
First we show that if X ∈ CM Λ satisfies projdim Λ X < ∞ , then projdim Λ X É n . By Lemma 3.8,if X ∈ CM Λ , then Ext i Λ ( X , Λ ) = i > n . Since Ext r Λ ( X , Λ ) r = projdim Λ X , we must have eitherprojdim Λ X É n or projdim Λ X = ∞ .Now, for any module X ∈ mod Λ with depth R X = t , the ( d − t ) th syzygy must be in CM Λ by the Depth Lemmafor R and the fact that Λ is MCM over R . We then haveprojdim Λ X = ( d − t ) + projdim Ω d − t X É d − t + n = dim R − depth R X + n .The upper bound of (3.1) follows at once from this. To prove the lower bound we simply note that projective Λ -modules are in CM Λ . By the Depth Lemma again, if depth R X = t , then the first syzygy which could beprojective is the ( d − t ) th , as each syzygy can go up in depth by at most 1. Thusprojdim Λ X Ê d − depth R X .This concludes the proof. (cid:3) It is well known that commutative rings of finite Krull dimension d have either infinite global dimension orfinite global dimension equal to d . Thus, for 1 É n < ∞ , we know commutative n -canonical orders cannot exist.The goal of the rest of this section is to give some examples of non-commutative n -canonical orders for n Ê Lemma 3.10.
Let Λ be an order over a Cohen-Macualay local ring ( R , m ) . Let M be a Λ -module and x ,... , x t ∈ m an M-regular sequence. Then we have projdim Λ ( M /( x ,... , x t ) M ) = projdim Λ M + t .6 roof. Consider the short exact sequence0 −→ M · x −−→ M −→ M / x M −→ i − Λ ( M , − ) −→ Ext i Λ ( M / x M , − ) −→ Ext i Λ ( M , − ) −→ Ext i Λ ( M , − ).From this we can deduce projdim Λ M / x M = projdim Λ M +
1. The result follows from a repeated application ofthis process. (cid:3)
The next result follows from Lemma 3.10 and Theorem 3.9 and is our first example of n -canonical orders. Corollary 3.11.
Let R be a d-dimensional CM local ring with a canonical module and Λ an R-order. We have gldim Λ Ê d. Moreover, if gldim Λ < ∞ , then Λ is an n-canonical order with n : = gldim Λ − d.Proof. The first claim is well-known. For the second claim, we assume that gldim Λ < ∞ . Since Λ is leftand right Noetherian, we know gldim Λ op = gldim Λ . Then, since ω Λ is in CM Λ op we can find an ω Λ -regularsequence x ,... , x d ∈ m , where m is the maximal ideal of R . Applying Lemma 3.10, we see thatprojdim Λ op ω Λ + d = projdim Λ op ( ω Λ /( x ,... , x n ) ω Λ ).Since projdim Λ op ( ω Λ /( x ,... , x n ) ω Λ ) É gldim Λ op = gldim Λ , we know projdim Λ op ω Λ É gldim Λ − d . (cid:3) Remark 3.12.
Note that the right inequality of Theorem 3.9 cannot be strengthened to an equality. For example,let Λ be a finite-dimensional algebra over a field such that gldim Λ = n > (many such algebras exist). Then, Λ is n-canonical by Corollary 3.11 and we have { projdim Λ X | X ∈ CM( Λ ) } = { n } . Example 3.13.
Let k be an infinite field and let R be the complete (2,1)-scroll, that is, R = k [[ x , y , z , u , v ]]/ I withI the ideal generated by the × minors of ¡ x y uy z v ¢ . Then, R is a 3-dimensional CM normal domain of finiteCM type [24, 16.12] . It is known Γ = End R ( R ⊕ ω ) is MCM over R. Moreover, Smith and Quarles have shown gldim( Γ ) = while dim R = . Thus Γ is a 1-canonical order. We now establish the existence of n -canonical orders for n Ê n -canonical property is additive under tensoring. Lemma 3.14.
Let Λ and Λ be algebras over a Gorenstein local ring R such that Λ and Λ are free R-modules.Then Λ ⊗ R Λ is an R-order and ω Λ ⊗ Λ ∼= ω Λ ⊗ R ω Λ as both Λ ⊗ R Λ -bimodules roof. This follows from the composition of Λ ⊗ R Λ -bimodule isomprhismsHom R ( Λ ⊗ R Λ , R ) ∼= Hom R ( Λ ,Hom R ( Λ , R )) ∼= Hom R ( Λ , R ) ⊗ R Hom R ( Λ , R ).Here, the first isomorphism is the canonical isomorphism of Hom-tensor adjunction. The second isomor-phism follows since Λ is a free R -module. (cid:3) Since we are now able to find the canonical module of orders which are tensor products of free R -modules,we get the following examples for R a regular local ring. Theorem 3.15.
Let ( R , m , k ) be a regular local ring. Suppose Λ , Λ are n -canonical and n -canonical R-orders, respectively. Then Λ ⊗ R Λ is an ( n + n ) -canonical R-order.Proof. Since Λ and Λ are MCM over R , and R is a regular local ring, then in fact they are free. Then,noting that ( Λ ⊗ R Λ ) op ∼= Λ op1 ⊗ R Λ op2 , the theorem follows immediately from Lemma 3.14 and [8, CorollaryIX.2.7], namely that projdim Λ op1 ⊗ R Λ op2 ( ω Λ ⊗ R Λ ) = projdim Λ op1 ⊗ R Λ op2 ( ω Λ ⊗ R ω Λ ) = projdim Λ op1 ω Λ + projdim Λ op2 ω Λ . (cid:3) With this in hand, we can prove the existence of orders which are n -canonical and have infinite globaldimension. We first remind the reader of some basics on path algebras. The main theorem of Chapter 3 is homological in nature.As such, we collect some background on the homological behavior of path algebras.
Definition 3.16. A quiver Q = ( Q , Q , s , t ) is a directed graph Q with vertex set Q and arrow set Q . Thereare two maps s , t : Q −→ Q where for an arrow a ∈ Q , s ( a ) is the origin of a and t ( a ) is the destination of a. A path in Q is a sequence of arrows a n a n − ... a such that t ( a i ) = s ( a i + ) for É i É n − For each vertex a ∈ Q ,we have the trivial path at a, denoted by e a , which is the path which begins at a, ends at a, and consists of noarrows. Definition 3.17.
Let R be a commutative Noetherian ring and Q a quiver. The path algebra
RQ of Q over Ris the free R-module with the basis the set of all paths a l a l − ... a of length l Ê in Q. The product of two basisvectors (i.e., paths) b k ... b and a l ... a of RQ is defined by ( b k ... b ) · ( a l ... a ) = b k ... b a l ... a f t ( a l ) = s ( b ) and 0 otherwise, i.e., the product of arrows b · a is nonzero if and only if b leaves the vertex wherea arrives. Multiplication is extended to linear combinations of basis elements R-linearly. The next result is well-known to experts, and it is what makes path algebras a convenient choice forrelating global dimension information about orders back to the commutative base rings; we include a prooffor convenience.
Proposition 3.18.
Let Q be an acyclic quiver. Let R be a regular local ring of dimension d and RQ the pathalgebra of Q over R. Then, gldim RQ É d + . If R is not regular, then gldim RQ = ∞ .Proof. Let R be a regular local ring with gldim R = d < ∞ . We note that the path algebra RQ is exactly thetensor algebra T S ( V ) where S : = RQ is the subring of RQ consisting only of linear combinations of the trivialpaths e i and V = RQ is an S -bimodule. Then, we have a canonical short exact sequence of R -modules0 −→ T S ( V ) ⊗ R V ⊗ R T S ( V ) f −→ T S ( V ) ⊗ R T S ( V ) mult . −−−−→ T S ( V ) −→ f is defined on basis elements by f ( x , v , y ) = xv ⊗ y − x ⊗ vy . To see that gldim T S ( V ) É gldim R +
1, weconsider a finitely generated left T S ( V )-module M which is a d th syzygy over T S ( V ). We can find such amodule over T S ( V ) since, by Corollary 3.11, we know gldim RQ Ê d . Such a module is necessarily projectiveover R since it is MCM over R by the Depth Lemma. It will suffice to show projdim T S ( V ) M É T S ( V ) ⊗ R T S ( V ) −→ T S ( V ) is split by the map g : T S ( V ) −→ T S ( V ) ⊗ R T S ( V ) givenby g ( v ) = ⊗ v , so the above exact sequence is split exact, and it remains exact when tensoring with M over T S ( V ). This yields an exact sequence of T S ( V )-modules0 −→ T S ( V ) ⊗ R V ⊗ R M −→ T S ( V ) ⊗ R M −→ M −→ V is projective over R by definition; M is projective over R by assumption; and extension ofscalars T S ( V ) ⊗ R − preserves projectivity. Hence, the above is a projective resolution of M over T S ( V ). Thusprojdim T S ( V ) M É
1. This verifies gldim RQ É d + RQ ∼= R ⊗ R RQ , and both R and RQ are free over R . Hence by[8, Corollary IX.2.7], if R has a module of infinite projective dimension (if R is local and non-regular, then theresidue field is such a module), then so has RQ . (cid:3) We record one more Lemma in order to work with path algebras efficiently.
Lemma 3.19.
Let R be an algebra over a commutative local ring T . Let Q a quiver, and I a right ideal in TQ.Then there is an isomorphism of R-algebrasRQ / I ( RQ ) ∼= ( TQ / I ) ⊗ T R .9 roof. We begin with the case I =
0. We regard TQ and R as subrings of RQ . Then, the multiplication givesa map Φ : TQ ⊗ T R −→ RQ . This map is clearly a morphism of R -algebras. Since both sides are free R -moduleswith basis given by all paths on Q , this is necessarily an isomprhism.We now move to the case that I is a non-zero ideal. We note R ⊗ T ( TQ / I ) ∼= R ⊗ T TQ ⊗ TQ TQ / I ∼= RQ ⊗ TQ TQ / I ∼= RQ / I ( RQ ),where the second isomorphism follows from the I = (cid:3) And now we can produce a natural example of n -canonical orders with infinite global dimension. Theorem 3.20.
Let ( R , m , k ) be a d-dimensional Gorenstein local domain. Suppose Q is an acyclic quiver. Then Λ = RQ is a 1-canonical R-order. If R is not regular, then gldim Λ = ∞ .For the proof we will need to reduce to the case where R is complete via the following lemma. Lemma 3.21.
Suppose R is a CM local ring with a canonical module ω R and that R , → S is a faithfully flat(commutative) ring extension such that dim S = dim R and S has a canonical module ω S = ( ω R ) ⊗ R S (e.g., ifS = b R). Let Λ be an R-order. We have that Λ is an n-canonical R-order if and only if Λ ⊗ R S is an n-canonicalS-order.Proof.
We must prove two facts. First we note that since S is faithfully flatHom R ( M , N ) ⊗ R S ∼= Hom S ( M ⊗ R S , N ⊗ R S ).It follows at once that ω Λ ⊗ R S ∼= ( ω Λ ) ⊗ R S . Verifying that this is a Λ ⊗ R S -isomorphism is straightforward. Next,since exactness of Λ -module sequences can be checked as R -modules, S is faithfully flat over R , and − ⊗ R S takes projective Λ -modules to projective Λ ⊗ R S -modules, we see thatprojdim Λ ω Λ = projdim Λ ⊗ R S ( ω Λ ⊗ R S ).The lemma follows at once from these two observations. (cid:3) Proof of Theorem 3.20.
We reduce to the case where R is complete. Let b R denote the completion of R withrespect to the maximal ideal. By Lemma 3.21, we see that RQ is 1-canonical if and only if RQ ⊗ R b R is 1-canonical. But, by Lemma 3.19, we know that RQ ⊗ b R ∼= b RQ . Thus we see RQ is 1-canonical if and only if b RQ is 1-canonical. Thus we may assume R is complete.Now, by Cohen’s Structure Theorem for complete local rings, [19, Theorem 8.24], R is an order oversome d -dimensional regular local ring S . Since R is a Gorenstein local ring and an order over S , we have R ∼= ω R ∼= Hom S ( R , S ) and projdim R ω R = R is MCM over S and hence free; i.e., R is a 0-canonical10 -order. Now, by Proposition 3.18, we know that gldim SQ É d +
1, and hence by Corollary 3.11 SQ is a 1-canonical S -order. Now, by Proposition 3.19 and Theorem 3.15, Λ : = RQ ∼= R ⊗ S SQ is a 1-canonical S -order.We observe that by Lemma 2.3, the canonical module of Λ as an S -order is the same as the canonical moduleof Λ as an R -order. Hence, Λ is a 1-canonical R -order. Lastly, by Lemma 3.18 we know that if R is not regular,we have gldim Λ = ∞ . (cid:3)
4. Higher Isolated Singularities.
The main theorem of this paper is that if an order Λ is n -canonical andhas only finitely many nonisomorphic indecomposable modules in Ω n CM Λ , then Λ has finite global dimensionon the punctured spectrum of R . In this section we show that over orders with this property, high syzygiesbehave much like MCM modules over isolated singularities. Definition 4.1.
Let Λ be an order over a CM ring R. We call Λ an n -isolated singularity if gldim Λ p É n + dim R p for all non-maximal prime ideals p . We say Λ is n -nonsingular if gldim Λ p É n + dim R p for all p ∈ Spec R. Remark 4.2.
It follows from the definition that if Λ is an n-isolated singularity, it is also an m-isolatedsingularity for any m Ê n. It might be interesting to study “strict” n-isolated singularities where gldim Λ p = n + dim R p for all p ∈ Spec
R, as well as non-strict ones.
When Λ is a isolated singularity (the n = Λ are d th syzygies; infact, the stronger statement that all modules in CM Λ are d -torsionfree Λ -modules is proved in [1, Theorem7.9]. This fact allows one to bound the projective dimension of modules in CM Λ on the punctured spectrum.Over n -isolated singularities, we accomplish this task with the following result. Lemma 4.3.
Let Λ be an n-isolated singularity over a CM local ring R. Then if M ∈ CM ( Λ ) we have projdim M p É nfor all non-maximal primes p .Proof. Let M ∈ CM Λ and p ∈ Spec R . It follows that M p ∈ CM Λ p . Pick a maximal M p -regular sequence x ,... , x t ∈ p R p . It follows from Lemma 3.10 thatprojdim Λ p M p /( x ,... , x t ) M p = projdim Λ p M p + dim R p .Since gldim Λ p É n + dim R p , it must be that projdim Λ p M p É n . (cid:3) n -isolated singularities. Recall that X n = add { Ω n CM Λ } . Lemma 4.4.
Let Λ be an order over a CM local ring R. Then Λ is an n-isolated singularity if and only if X p isa projective Λ p -module for all X ∈ X n and non-maximal primes p ∈ Spec
R.Proof. ( ⇒ ): This follows at once from the previous lemma.( ⇐ ): Fix a non-maximal prime p ∈ Spec R . We must show gldim Λ p É n + dim R p . Let X ∈ mod Λ p . We know,then, that X = M p for some M ∈ mod Λ . We note that Ω n + d Λ M ∈ X n , so Ω n + d Λ p X = ( Ω n + d Λ M ) p is a projective Λ -module. Hence, projdim Λ p X < ∞ . Thus, we have established that gldim Λ p < ∞ . Since ω Λ ∈ CM Λ , it is clearthat Ω n ( ω Λ ) ∈ X n . Hence Ω n ( ω Λ p ) = ( Ω n ( ω Λ )) p is projective by assumption. Thus, projdim Λ p ω Λ p É n and Λ op p is n -canonical. Since gldim Λ op p < ∞ , Proposition 3.6 implies that Λ p is n -canonical so projdim Λ op p ω Λ p É n andTheorem 3.9 tells us gldim Λ p É n + dim R p . (cid:3) The following lemma will be useful later, as it detects n -isolated singularities. Lemma 4.5.
Let R be a CM local ring with canonical module ω . Let Λ be an R-order. Then Λ is an n-isolatedsingularity if and only if ℓ R (Ext Λ ( N , M )) < ∞ for all M , N ∈ X n .Proof. ( ⇒ ): This follows at once from Lemma 4.3.( ⇐ ): Suppose ℓ (Ext Λ ( N , M )) < ∞ for all M , N ∈ X n . Let p be a prime ideal of R which is not maximal.Consider a module X ∈ mod Λ p . We know X = M p for some M ∈ mod Λ . Consider the exact sequence over Λ p ,(4.1) 0 −→ Ω n + d + ( X ) −→ F −→ Ω n + d ( X ) −→ F is a free Λ p -module. This is the same as the exact sequence(4.2) 0 −→ Ω n + d + Λ p ( M p ) −→ F −→ Ω n + d Λ p ( M p ) −→ Ω i ( X ) = ( Ω i M ) p for all i Ê Ω n + d M , Ω n + d + M ∈ X n , it follows thatExt Λ p ( Ω n + d X , Ω n + d + X ) ∼= ³ Ext Λ ( Ω n + d M , Ω n + d + M ) ´ p = Λ ( Ω n + d M , Ω n + d + M ) has finite length by assumption. This meanssequence (4.1) splits, and hence Ω n + d X is Λ p -projective, and, therefore, projdim Λ p X < ∞ and so gldim Λ p < ∞ .We can apply a similar argument as above to X = ω Λ p = ( ω Λ ) p to see that projdim Λ p ω Λ p É n . In this case, weonly need to consider Ω n X since we already have ω Λ ∈ CM Λ . Thus we conclude Λ opp is n -canonical. Anidentical argument to the end of the proof of Lemma 4.4 gives that gldim Λ p É n + dim R p . (cid:3) n th syzygies (of MCM modules) over an n -isolated singularity behavelike MCM modules over an isolated singularity. This is shown for the n = Proposition 4.6.
Let Λ be an n-isolated singularity over a d-dimensional CM local ring R. For X ∈ X n :(1) Ext i Λ op (Tr X , Λ ) = for i = d.(2) Ext i Λ ( X , Y ) , Tor Λ i ( Z , X ) , and Hom Λ ( X , Y ) are all finite length for any Y ∈ mod Λ and Z ∈ mod Λ op .Proof. Let p ∈ Spec R be non-maximal. We see if X ∈ X n , then X p is projective over Λ p by Lemma 4.3, andthus (2) holds. For assertion (1), we note that if d = d = X isprojective on the punctured spectrum, by Lemma 4.4. This implies that Ext Λ op (Tr X , Λ ) has finite length sinceTr X p = p . Then the well-known exact sequence (see, e.g., [18, Proposition12.8]) 0 −→ Ext Λ op (Tr X , Λ ) −→ X −→ X ∗∗ −→ Ext Λ op (Tr X , Λ ) −→ Λ op (Tr X , Λ ) embeds in X . But, depth R X Ê d Ê X cannot contain a module of depthzero. Thus, Ext Λ op (Tr X , Λ ) = i Λ op (Tr X , Λ ) = i = k − É k É d . We begin with a projective resolution... −→ P k −→ P k − −→ ... −→ P −→ P −→ Tr X −→ X ∼= X ∗∗ , we get an exact sequence0 −→ X −→ P ∗ −→ P ∗ −→ ... −→ P ∗ k − −→ ( Ω k Tr X ) ∗ −→ Ext k Λ op (Tr X , Λ ) −→ R ( Ω k Tr X ) ∗ Ê
2. Since Ext k Λ op (Tr X , Λ ) has finite length and P ∗ i ∈ CM Λ for all i , the Depth Lemmaimplies depth R X É d −
1, which is impossible since X ∈ CM Λ . Thus, it must be that Ext k Λ op (Tr X , Λ ) =
0. Thuspart (1) is proved by induction. (cid:3)
The following is the analog of [15, Prop 2.17], and the proof is similar.
Proposition 4.7.
Let Λ be an order over a CM ring R of Krull dimension d with canonical module ω R . Thefollowing are equivalent:(1) Λ is n-nonsingular.(2) gldim Λ m É n + d for all maximal ideals m ∈ Spec
R.(3) CM Λ ⊂ projdim É n Λ .(4) projdim Λ op ω Λ É n and gldim Λ < ∞ . roof. The first 3 implications are the same argument as [15], but we include them for the convenience ofthe reader. (1) ⇒ (2) This is immediate.(2) ⇒ (3) This proof is nearly identical to the proof of Lemma 4.3.(3) ⇒ (4) Since ω Λ ∈ CM Λ , we know it has projective dimension over Λ at most n by (3). Also, since each d th syzygy is MCM by the Depth Lemma, we have gldim Λ < ∞ . Since gldim Λ op = gldim Λ < ∞ , Proposition 3.6then implies that projdim Λ op ω Λ É n .(4) ⇒ (1) Let X be in CM( Λ p ). Since localization can only reduce projective dimension, we have thatprojdim Λ op p ω Λ p É n and gldim Λ p < ∞ . The result then follows from Theorem 3.9 (cid:3) Remark 4.8.
One might ask if we can strengthen condition (3) to be a set equality. If n Ê , the answer is no:consider a regular sequence x = x ,... , x d on Λ , and take the Koszul complex over Λ on x. Then this is exact andhas length d. Then Ω d − ( Λ / x Λ ) has depth d − by the Depth Lemma, but the end of the Koszul complex givesa length one resolution. Thus Ω d − ( Λ / x Λ ) ∈ projdim É n Λ but is not in CM Λ . It is clear that (3) is equivalent to X n ⊂ proj Λ .
5. Gorenstein Projectives and Auslander’s Theorem.
The goal of this section is to prove the followingvariation of Auslander’s Theorem, [2].
Theorem 5.1.
Let R be a CM local ring with canonical module and suppose Λ is an R-order such that Λ and Λ op are n-canonical. If Λ has only finitely many nonisomorphic indecomposable modules in S , then Λ is ann-isolated singularity. The proof of this will rely on the notion of
Gorenstein Projective modules. Originally defined by Auslanderand Bridger in [3], a module M over an order Λ is called Gorenstein Projective if M is reflexive (i.e., the naturalmap M −→ M ∗∗ is an isomorphism) andExt i Λ ( M , Λ ) = Ext i Λ ( M ∗ , Λ ) = i > G proj Λ denote the full subcategory of mod Λ consisting of all Gorenstein projective modules. Ourinterest in Gorenstein projectives is motivated by the following fact, which is a variation of a well-knownproperty for n -Gorenstein rings. For details, see [10, Section 10.2]. Proposition 5.2.
Let R be a CM local ring with canonical module ω . Suppose Λ is an R-order suchthat Λ and Λ op are n-canonical, where n Ê . Let M be a Λ -module. Then, M ∈ G proj if and only ifM ∈ X n = add { Ω n CM( Λ ) } . roof. Since Gorenstein projectives occur as syzygies in complete resolutions, it is clear that G proj Λ ⊂ add Ω n CM Λ .To complete the proof we show the reverse inclusion. Let M = Ω n X for a maximal Cohen-Macaulay module X ,and suppose M is not a projective module. By Lemma 3.8 we have that Ext i Λ ( M , Λ ) = i >
0. Then, bydualizing a projective resolution of M , we get an exact sequence0 −→ M ∗ −→ P ∗ −→ P ∗ −→ ··· .According to Lemma 4.3, M necessarily has infinite projective dimension if it is not projective; therefore, wesee M ∗ is an arbitrarily high syzygy. By Lemma 3.8 again we have Ext i Λ op ( M ∗ , Λ ) = i >
0. All that remainsto show is that M is reflexive. Note that Tr M fits into the above exact sequence as follows0 −→ Tr M −→ P ∗ −→ P ∗ −→ ··· .Thus, Tr M is also an arbitrarily high syzygy and satisfies the same Ext vanishing as M . Thus the exactsequence 0 −→ Ext Λ op (Tr M , Λ ) −→ M −→ M ∗∗ −→ Ext Λ op (Tr M , Λ ) −→ M ∼= M ∗∗ . (cid:3) The key use of Gorenstein projectives is that they are closed under extensions. This has been shown invarious places, see e.g., [5, Proposition 5.1].
Corollary 5.3.
Let Λ be an order over a CM local ring R with a canonical module. Suppose Λ and Λ op aren-canonical. Then X n is closed under extensions. We now return to proving the main theorem. The proof of this involves several lemmas, and it followsclosely Huneke and Leuschke’s proof of Auslander’s Theorem, [13]. The following Theorem due to Miyata isour first step.
Lemma 5.4. [20, Theorem 2]
Let Λ be a module finite algebra over a commutative Noetherian ring R. Supposewe have an exact sequence of finitely generated Λ -modulesM −→ X −→ N −→ and that X ∼= M ⊕ N. Then the sequence is a split short exact sequence.
From this we are able to prove the following lemma about Ext Λ ( N , M ). The proof is similar to the one in[13]; it is omitted for this reason. 15 emma 5.5. Let ( R , m ) be a CM local ring and Λ an R-order. Fix r ∈ m . Suppose we have an exact sequence of Λ -modules, α : 0 −→ M −→ X α −→ N −→ and a commutative diagram α : 0 −−−−−→ M −−−−−→ X α −−−−−→ N −−−−−→ r y f y °°° r α : 0 −−−−−→ M −−−−−→ X r α −−−−−→ N −−−−−→ If X α ∼= X r α , then α ∈ r Ext Λ ( N , M ) . Now, we are able to prove the following lemma from which the main theorem follows. The proof is a straight-forward generalization of the commutative case.
Lemma 5.6.
Suppose Λ is an order over a CM local ring ( R , m , k ) . Given Λ -modules M and N, if there are onlyfinitely many choices (up to isomorphism) for X such that there is an exact sequence of Λ -modules −→ M −→ X −→ N −→ then Ext i Λ ( N , M ) is a finite length R-module.Proof. Let α ∈ Ext Λ ( N , M ) and r ∈ m . It is well known that an R -module M has finite length if and only iffor all r ∈ m and x ∈ M there is an integer n so that r n x =
0. Thus, we must only show that r n α = n ≫ n we consider a representative r n α : 0 −→ M −→ X n −→ N −→ X n can exist up to isomorphism there is an infinite sequence n < n < n < ... suchthat X n i ∼= X n j for all pairs i , j . Set β = r n α , and let i >
1. Then r n i α = r n i − n β . We show β =
0. We have, foreach i , a commutative diagram β : 0 −−−−−→ M −−−−−→ X n −−−−−→ N −−−−−→ r ni − n y y °°° | r n i − n β : 0 −−−−−→ M −−−−−→ X n i −−−−−→ N −−−−−→ X n ∼= X n i , we have β ∈ r n i − n Ext Λ ( N , M ) for every i . Since the sequence of n i isinfinite and strictly increasing, this means β ∈ m t Ext Λ ( N , M ) for all t . Finally, the Krull Intersection Theorem[19, Theorem 8.10] implies β = (cid:3) X n = add { Ω n CM Λ } . Proof of Theorem 5.1.
Let M , N ∈ X n . By Lemma 4.5 we must only show that ℓ R (Ext Λ ( N , M )) < ∞ . Considerany sequence α ∈ Ext Λ ( N , M ), α : 0 −→ M −→ X −→ N −→ X ∈ X n . Now since M and N are finitely generated and there are only finitely manyindecomposable modules in X n , there are only finitely many possibilities for X . Namely, X must be one of thefinitely many modules in X n generated by at most µ Λ ( M ) + µ Λ ( N ), where µ Λ ( Y ) denotes the minimum numberof generators of Y over Λ . Thus, ℓ R (Ext Λ ( N , M )) < ∞ by Lemma 5.6. (cid:3)
6. Application to Commutative Rings.
In view of Theorem 5.1 and 3.20, we arrive at the followinggeneralization of Auslander’s Theorem in the case where R is a suitable Gorenstein local ring. Corollary 6.1.
Let R be a Gorenstein local ring which is an order over a regular local ring S (e.g., if Ris complete), and let Q be an acyclic quiver. If there exist only finitely many nonisomorphic indecomposablemodules in Ω CM( RQ ) , then R is an isolated singularity, i.e., gldim R p = dim R p for all non-maximal primes ideals p ∈ Spec
R.Proof.
We notice that by Theorem 3.20 RQ is a 1-canonical order. Additionally, it is clear that RQ op isobtained by taking the path algebra over R of the quiver obtained by reversing all arrows in Q . It followsthat RQ op is also a 1-canonical R -order. Thus, by Theorem 5.1, if there are only finitely many indecomposablemodules in Ω CM( RQ ) we must have that RQ is a 1-isolated singularity. By Proposition 3.18, we know for anycommutative ring R that gldim RQ < ∞ if and only if gldim R < ∞ . Thus, RQ can be a 1-isolated singularityif and only if gldim R p < ∞ for all non-maximal primes p . Since R is commutative, this is only possible ifgldim R p = dim R p . (cid:3) We note that the proof of Corollary 6.1 does not require completeness beyond ensuring R is an order over aregular local ring. It would be nice to remove this assumption. In this vein we have the following question: Question 6.2.
For a local ring R and an acyclic quiver Q, is it true that RQ has only finitely manyindecomposable modules in X n if and only if d RQ ∼= b RQ has?
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